Lemma 63.10.2. Let f : X \to Y be a finite type separated morphism of quasi-compact and quasi-separated schemes. Let \Lambda be a ring. The functors Rf_! constructed in Section 63.9 are bounded in the following sense: There exists an integer N such that for E \in D^+_{tors}(X_{\acute{e}tale}, \Lambda ) or E \in D(X_{\acute{e}tale}, \Lambda ) if \Lambda is torsion, we have
H^ i(Rf_!(\tau _{\leq a}E) \to H^ i(Rf_!(E)) is an isomorphism for i \leq a,
H^ i(Rf_!(E)) \to H^ i(Rf_!(\tau _{\geq b - N}E)) is an isomorphism for i \geq b,
if H^ i(E) = 0 for i \not\in [a, b] for some -\infty \leq a \leq b \leq \infty , then H^ i(Rf_!(E)) = 0 for i \not\in [a, b + N].
Proof.
Assume \Lambda is torsion and consider the functor Rf_! : D(X_{\acute{e}tale}, \Lambda ) \to D(Y_{\acute{e}tale}, \Lambda ). By construction it suffices to prove this when f is an open immersion and when f is a proper morphism. For any open immersion j : U \to X of schemes, the functor j_! : D(U_{\acute{e}tale}) \to D(X_{\acute{e}tale}) is exact and hence the statement holds with N = 0 in this case. If f is proper then Rf_! = Rf_*, i.e., it is a right derived functor. Hence the bound on the left by Derived Categories, Lemma 13.16.1. Moreover in this case f_* : \textit{Mod}(X_{\acute{e}tale}, \Lambda ) \to \textit{Mod}(Y_{\acute{e}tale}, \Lambda ) has bounded cohomological dimension by Morphisms, Lemma 29.28.5 and Étale Cohomology, Lemma 59.92.2. Thus we conclude by Derived Categories, Lemma 13.32.2.
Next, assume \Lambda is arbitrary and let us consider the functor Rf_! : D^+_{tors}(X_{\acute{e}tale}, \Lambda ) \to D^+_{tors}(Y_{\acute{e}tale}, \Lambda ). Again we immediately reduce to the case where f is proper and Rf_! = Rf_*. Again part (1) is immediate. To show part (3) we can use induction on b - a, the distinguished triangles of trunctions, and Étale Cohomology, Lemma 59.92.2. Part (2) follows from (3). Details omitted.
\square
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